Vertex operator algebras and the Verlinde conjecture

We prove the Verlinde conjecture in the following general form: Let V be a simple vertex operator algebra satisfying the following conditions: (i) The homogeneous subspaces of V of weights less than 0 are 0, the homogeneous subspace of V of weight 0 is spanned by the vacuum and V' is isomorphic to V as a V-module. (ii) Every weak V-module gradable by nonnegative integers is completely reducible. (iii) V is C_2-cofinite. (In the presence of Condition (i), Conditions (ii) and (iii) are equivalent to a single condition, namely, that every weak V-module is completely reducible.) Then the matrices formed by the fusion rules among the irreducible V-modules are diagonalized by the matrix given by the action of the modular transformation \tau\mapsto -1/\tau on the space of characters of irreducible V-modules. Using this result, we obtain the Verlinde formula for the fusion rules. We also prove that the matrix associated to the modular transformation \tau\mapsto -1/\tau is symmetric.Comment: 62 pages. To appear in Communications in Contemporary Mathematics. Referee's comments have been taken into accoun

Simplified preferences, voting, and the power of combination.

In this paper we interpreted the decision to vote for a particular party as a process of delegation to decision makers having a simplified system of preferences. Each person in a population votes for the political party that place priority on one or more issues that they consider important. Moreover, on the basis of a survey on preferences of population, we have simulated a delegation procedure which chart the selection process of a particular party. Finally, making use of noncommutative harmonic analysis, we decomposed the delegation function, and isolated the effect of a particular affinity, or a combination of either the pair of items that characterize a party. We used noncommutative harmonic analysis as an application of some results obtained by Michael E. Orrison and Brian L. %%@ Lawson in relation to spectral analysis applied in voting in political committees.

Monodromy--like Relations for Finite Loop Amplitudes

We investigate the existence of relations for finite one-loop amplitudes in Yang-Mills theory. Using a diagrammatic formalism and a remarkable connection between tree and loop level, we deduce sequences of amplitude relations for any number of external legs.Comment: 24 pages, 6 figures, v2 typos corrected, reference adde

External leg corrections in the unitarity method

Unitarity cuts diverge in the channel of a single massive external fermion. We propose an off-shell continuation of the momentum that allows a finite evaluation of the unitarity cuts. If the cut is taken with complete amplitudes on each side, our continuation and expansion around the on-shell configuration produces the finite contribution to the bubble coefficient. Finite parts in the expansion of the external leg counterterms must be included explicitly as well.Comment: 28 pages, 9 figures. Published version. Eq. (B.17) corrected, minor clarifications, typos fixe

Grid multi-category response logistic models.

BackgroundMulti-category response models are very important complements to binary logistic models in medical decision-making. Decomposing model construction by aggregating computation developed at different sites is necessary when data cannot be moved outside institutions due to privacy or other concerns. Such decomposition makes it possible to conduct grid computing to protect the privacy of individual observations.MethodsThis paper proposes two grid multi-category response models for ordinal and multinomial logistic regressions. Grid computation to test model assumptions is also developed for these two types of models. In addition, we present grid methods for goodness-of-fit assessment and for classification performance evaluation.ResultsSimulation results show that the grid models produce the same results as those obtained from corresponding centralized models, demonstrating that it is possible to build models using multi-center data without losing accuracy or transmitting observation-level data. Two real data sets are used to evaluate the performance of our proposed grid models.ConclusionsThe grid fitting method offers a practical solution for resolving privacy and other issues caused by pooling all data in a central site. The proposed method is applicable for various likelihood estimation problems, including other generalized linear models

Towards the Amplituhedron Volume

21 pages; v2: version published in JHEPIt has been recently conjectured that scattering amplitudes in planar N=4 super Yang-Mills are given by the volume of the (dual) amplituhedron. In this paper we show some interesting connections between the tree-level amplituhedron and a special class of differential equations. In particular we demonstrate how the amplituhedron volume for NMHV amplitudes is determined by these differential equations. The new formulation allows for a straightforward geometric description, without any reference to triangulations. Finally we discuss possible implications for volumes related to generic N^kMHV amplitudes.Peer reviewe

Multiresolution Analysis of Long Time Series with Applications to Finance

We consider multi-resolution time series models and their application to high-frequency financial data. An individual transaction share price of a specific firm is subject to market microstructure noise. Therefore, we propose trading duration time weighted averages over given time intervals. Averages over long intervals lead to a coarse resolution and averaging over shorter intervals lead to a finer resolution. Arranging sub-intervals of given lengths on scales with coarse to fine resolution imply a structure which can be represented as a directed acyclic graph. Time series models are then formulated using this graph structure. It is shown that these models have a linear state space representation which allows for efficient computation of the likelihood needed in parameter estimation and for a straightforward treatment of missing observations. Application of these models to the log transaction prices of the IBM shares traded at the New York Stock Exchange from February until October 2002 show that the corresponding one-step prediction errors are heavy tailed and therefore a specific variance term is allowed to follow a fiEGARCH specification, improving the tail behavior and leading to a better fit